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- I CAN identify and generate geometric sequences and relate them to exponential functions.

A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a nonzero constant r called the common ratio.

16 – 8 = 8

24 – 16 = 8

32 – 24 = 8

Identify Geometric Sequences

A. Determine whether each sequence is arithmetic, geometric, or neither. Explain.

0, 8, 16, 24, 32, ...

0 8 16 24 32

Answer: The common difference is 8. So the sequence is arithmetic.

Example 13

3

__

__

__

27

36

48

3

__

___

___

___

=

=

=

4

4

4

4

64

48

36

Answer: The common ratio is , so the sequence is geometric.

Identify Geometric Sequences

B. Determine whether each sequence is arithmetic, geometric, or neither. Explain.

64, 48, 36, 27, ...

64 48 36 27

Example 1B

C

A. Determine whether the sequence is arithmetic, geometric, or neither.1, 7, 49, 343, ...

A. arithmetic

B. geometric

C. neither

Example 1B

C

B. Determine whether the sequence is arithmetic, geometric, or neither.1, 2, 4, 14, 54, ...

A. arithmetic

B. geometric

C. neither

Example 1______

64

64

___

= –8

= –8

= –8

–8

__

–8

1

Find Terms of Geometric Sequences

A. Find the next three terms in the geometric sequence.

1, –8, 64, –512, ...

Step 1 Find the common ratio.

1 –8 64 –512

The common ratio is –8.

Example 2× (–8)

× (–8)

× (–8)

Find Terms of Geometric Sequences

Step 2 Multiply each term by the common ratio to find the next three terms.

–512

4096

–32,768

262,144

Answer: The next 3 terms in the sequence are 4096; –32,768; and 262,144.

Example 21

1

1

__

__

__

__

5

20

10

___

___

___

2

2

2

2

40

20

10

=

=

=

The common ratio is .

Find Terms of Geometric Sequences

B. Find the next three terms in the geometric sequence.

40, 20, 10, 5, ....

Step 1 Find the common ratio.

40 20 10 5

Example 2×

×

1

5

1

5

5

5

5

5

1

__

__

__

__

__

__

__

__

__

8

2

2

4

2

8

2

4

2

Answer: The next 3 terms in the sequence are ,

, and .

Find Terms of Geometric Sequences

Step 2 Multiply each term by the common ratio to find the next three terms.

5

Example 2B

C

D

A. Find the next three terms in the geometric sequence.1, –5, 25, –125, ....

A. 250, –500, 1000

B. 150, –175, 200

C. –250, 500, –1000

D. 625, –3125, 15,625

Example 2B

C

D

B. Find the next three terms in the geometric sequence.800, 200, 50, , ....

A. 15, 10, 5

B. , ,

C. 12, 3,

D. 0, –25, –50

25

____

3

__

25

___

25

25

128

__

__

4

32

2

8

Example 24

–2

___

___

___

1

–2

4

= –2

= –2

= –2

Find the nth Term of a Geometric Sequence

A. Write an equation for the nth term of the geometric sequence 1, –2, 4, –8, ... .

The first term of the sequence is 1. So, a1 = 1. Now find the common ratio.

1 –2 4 –8

The common ration is –2.

an = a1rn – 1 Formula for the nth term

an = 1(–2)n – 1a1 = 1 and r = –2

Answer:an = 1(–2)n – 1

Example 3Find the nth Term of a Geometric Sequence

B. Find the 12th term of the sequence.1, –2, 4, –8, ... .

an = a1rn – 1 Formula for the nth term

a12 = 1(–2)12 – 1 For the nth term, n = 12.

= 1(–2)11 Simplify.

= 1(–2048) (–2)11 = –2048

= –2048 Multiply.

Answer: The 12th term of the sequence is –2048.

Example 3B

C

D

A.an = 3(–4)n – 1

B.an = 3( )n – 1

C.an = 3( )n – 1

D.an = 4(–3)n – 1

1

1

__

__

4

3

A. Write an equation for the nth term of the geometric sequence 3, –12, 48, –192, ....

Example 3B

C

D

B. Find the 7th term of this sequence using the equation an = 3(–4)n – 1.

A. 768

B. –3072

C. 12,288

D. –49,152

Example 3ART A 50-pound ice sculpture is melting at a rate in which 80% of its weight remains each hour. Draw a graph to represent how many pounds of the sculpture is left at each hour.

Compared to each previous hour, 80% of the weight remains. So, r = 0.80. Therefore, the geometric sequence that models this situation is 50, 40, 32, 25.6, 20.48,… So after 1 hour, the sculpture weighs 40 pounds, 32 pounds after 2 hours, 25.6 pounds after 3 hours, and so forth. Use this information to draw a graph.

Example 4B

C

D

A.B.

C.D.

SoccerA soccer tournament begins with 32 teams in the first round. In each of the following rounds, on half of the teams are left to compete, until only one team remains. Draw a graph to represent how many teams are left to compete in each round.

Example 4
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